In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. Mathematics is the study of quantity, structure, space and change. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

The definition of kernel takes various forms in various contexts. But in all of them, the kernel of a homomorphism is trivial (in a sense relevant to that context) if and only if the homomorphism is injective. The fundamental theorem on homomorphisms (or first isomorphism theorem) is a theorem, again taking various forms, that applies to the quotient algebra defined by the kernel. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ... In abstract algebra, for a number of algebraic structures, the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. ... In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

In this article, we first survey kernels for some important types of algebraic structures; then we give general definitions from universal algebra for generic algebraic structures. In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ...

Survey of examples

Linear operators

Let V and W be vector spaces and let T be a linear transformation from V to W. If 0_W is the zero vector of W, then the kernel of T is the preimage of the singleton set {0_W}; that is, the subset of V consisting of all those elements of V that are mapped by T to the element 0_W. The kernel is usually denoted "ker T" (or a variation). In symbols: A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, a singleton is a set with exactly one element. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

Since a linear transformation preserves zero vectors, the zero vector 0_V of V must belong to the kernel. The transformation T is injective if and only if its kernel is only the singleton set {0_V}.

It turns out that ker T is always a subspace of V. Thus, it makes sense to speak of the quotient space V/(ker T). The first isomorphism theorem for vector spaces states that this quotient space is naturally isomorphic to the image of T (which is a subspace of W). As a consequence, the dimension of V equals the dimension of the kernel plus the dimension of the image. The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by collapsing N to zero. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... Image of the Wikimedia Commons logo. ... In mathematics, the dimension of a vector space V is the cardinality (i. ...

If V and W are finite-dimensional and bases have been chosen, then T can be described by a matrix M, and the kernel can be computed by solving the homogenous system of linear equations Mv = 0. In this representation, the kernel corresponds to the nullspace of M. The dimension of the nullspace, called the nullity of M, is given by the number of columns of M minus the rank of M, as a consequence of the rank-nullity theorem. In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... For the square matrix section, see square matrix. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ... The null space (also nullspace) of a matrix A is the set of all vectors v which solve the equation Av = 0, a linear subspace of the space of all vectors. ... In linear algebra, the nullity of a matrix M is the number of columns of M minus the rank of M. If the m by n matrix M is regarded as a linear transformation Rn → Rm, then the nullity is equal to the dimension of the kernel of this linear... In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... In mathematics, the rank-nullity theorem of linear algebra, in its simplest form, relates the rank and the nullity of a matrix together with the number of columns of the matrix. ...

Solving homogeneous differential equations often amounts to computing the kernel of certain differential operators. For instance, in order to find all twice-differentiable functions f from the real line to itself such that Homogeneous is an adjective that has several meanings. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, the real line is simply the set of real numbers. ...

xf''(x) + 3f'(x) = f(x),

let V be the space of all twice differentiable functions, let W be the space of all functions, and define a linear operator T from V to W by

(Tf)(x) = xf''(x) + 3f'(x) - f(x)

for f in V and x an arbitrary real number. Then all solutions to the differential equation are in ker T. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

One can define kernels for homomorphisms between modules over a ring in an analogous manner. This includes kernels for homomorphisms between abelian groups as a special case. This example captures the essence of kernels in general abelian categories; see Kernel (category theory). In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. ... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ...

Group homomorphisms

Let G and H be groups and let f be a group homomorphism from G to H. If e_H is the identity element of H, then the kernel of f is the preimage of the singleton set {e_H}; that is, the subset of G consisting of all those elements of G that are mapped by f to the element e_H. The kernel is usually denoted "ker f" (or a variation). In symbols: In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

Since a group homomorphism preserves identity elements, the identity element e_G of G must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {e_G}.

It turns out that ker f is not only a subgroup of G but in fact a normal subgroup. Thus, it makes sense to speak of the quotient group G/(ker f). The first isomorphism theorem for groups states that this quotient group is naturally isomorphic to the image of f (which is a subgroup of H). In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... Image of the Wikimedia Commons logo. ...

In the special case of abelian groups, this works in exactly the same way as in the previous section. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

Ring homomorphisms

Let R and S be rings and let f be a ring homomorphism from R to S. If 0_S is the zero element of S, then the kernel of f is the preimage of the singleton set {0_S}; that is, the subset of R consisting of all those elements of R that are mapped by f to the element 0_S. The kernel is usually denoted "ker f" (or a variation). In symbols: In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

Since a ring homomorphism preserves zero elements, the zero element 0_R of R must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {0_R}.

It turns out that, although ker f is generally not a subring of R since it may not contain the multiplicative identity, it is nevertheless a two-sided ideal of R. Thus, it makes sense to speak of the quotient ring R/(ker f). The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of f (which is a subring of S). Given a ring (R, +, *), we say that a subset S of R is a subring thereof if it is a ring under the restriction of + and * thereto, and contains the same unity as R. A subring is just a subgroup of (R, +) which contains the identity and is closed under... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers like even number or multiple of 3. For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... Image of the Wikimedia Commons logo. ...

To some extent, this can be thought of as a special case of the situation for modules, since these are all bimodules over a ring R: In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. ...

• R itself;
• any two-sided ideal of R (such as ker f);
• any quotient ring of R (such as R/(ker f)); and
• the codomain of any ring homomorphism whose domain is R (such as S, the codomain of f).

However, the isomorphism theorem gives a stronger result, because ring isomorphisms preserve multiplication while module isomorphisms (even between rings) in general do not. A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ...

This example captures the essence of kernels in general Mal'cev algebras.

Monoid homomorphisms

Let M and N be monoids and let f be a monoid homomorphism from M to N. Then the kernel of f is the subset of the direct product M × M consisting of all those ordered pairs of elements of M whose components are both mapped by f to the same element in N. The kernel is usually denoted "ker f" (or a variation). In symbols: In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...

Since f is a function, the elements of the form (m,m) must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the diagonal set {(m,m) : m in M}. In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

It turns out that ker f is an equivalence relation on M, and in fact a congruence relation. Thus, it makes sense to speak of the quotient monoid M/(ker f). The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of f (which is a submonoid of N). In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... Image of the Wikimedia Commons logo. ...

This is very different in flavour from the above examples. In particular, the preimage of the identity element of N is not enough to determine the kernel of f. This is because monoids are not Mal'cev algebras.

Universal algebra

All the above cases may be unified and generalized in universal algebra. Universal algebra is the field of mathematics that studies the ideas common to all algebraic structures. ...

General case

Let A and B be algebraic structures of a given type and let f be a homomorphism of that type from A to B. Then the kernel of f is the subset of the direct product A × A consisting of all those ordered pairs of elements of A whose components are both mapped by f to the same element in B. The kernel is usually denoted "ker f" (or a variation). In symbols: In abstract algebra, an algebraic structure consists of a set together with a collection of operations or relations defined on it which satisfy certain axioms. ... In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...

Since f is a function, the elements of the form (a,a) must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the diagonal set {(a,a) : a in A}. In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

It turns out that ker f is an equivalence relation on A, and in fact a congruence relation. Thus, it makes sense to speak of the quotient algebra A/(ker f). The first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B). In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... Image of the Wikimedia Commons logo. ... In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the...

Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely set-theoretic concept. For more on this general concept, outside of abstract algebra, see kernel of a function. In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... In mathematics, the kernel of a function f may be taken to be either the equivalence relation on the functions domain that roughly expresses the idea of equivalent as far as the function f can tell, or the corresponding partition of the domain. ...

Mal'cev algebras

In the case of Mal'cev algebras, this construction can be simplified. Every Mal'cev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of groups, and the zero element in the case of rings or modules). The characteristic feature of a Mal'cev algebra is that we can recover the entire equivalence relation ker f from the equivalence class of the neutral element. In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x in X | x ~ a } The notion of equivalence classes is useful for constructing sets...

To be specific, let A and B be Mal'cev algebraic structures of a given type and let f be a homomorphism of that type from A to B. If e_B is the neutral element of B, then the kernel of f is the preimage of the singleton set {e_B}; that is, the subset of A consisting of all those elements of A that are mapped by f to the element e_B. The kernel is usually denoted "ker f" (or a variation). In symbols: In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, a singleton is a set with exactly one element. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

Since a Mal'cev algebra homomorphism preserves neutral elements, the identity element e_A of A must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {e_A}.

The notion of ideal generalises to any Mal'cev algebra (as subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ring ideal in the case of rings, and submodule in the case of modules). It turns out that although ker f may not be a subalgebra of A, it is nevertheless an ideal. Then it makes sense to speak of the quotient algebra G/(ker f). The first isomorphism theorem for Mal'cev algebras states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B). The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In abstract algebra, a module is a generalization of a vector space. ... In abstract algebra, a module is a generalization of a vector space. ... In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

The connection between this and the congruence relation is for more general types of algebras is as follows. First, the kernel-as-an-ideal is the equivalence class of the neutral element e_A under the kernel-as-a-congruence. For the converse direction, we need the notion of quotient in the Mal'cev algebra (which is division on either side for groups and subtraction for vector spaces, modules, and rings). Using this, elements a and a' of A are equivalent under the kernel-as-a-congruence if and only if their quotient a/a' is an element of the kernel-as-an-ideal. In mathematics, a quotient is the end result of a division problem. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the reverse operation of multiplication, and sometimes it can be interpreted as repeated subtraction. ... In mathematics, subtraction is one of the four basic arithmetic operations. ...

Abelian algebras

I need to look up more stuff on universal algebra.

Algebras with nonalgebraic structure

Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations. For example, one may consider topological groups or topological vector spaces, with are equipped with a topology. In this case, we would expect the homomorphism f to preserve this additional structure; in the topological examples, we would want f to be a continuous map. The process may run into a snag with the quotient algebras, which may not be well-behaved. In the topological examples, we can avoid problems by requiring that topological algebraic structures be Hausdorff (as is usually done); then the kernel (however it is constructed) will be a closed set and the quotient space will work fine (and also be Hausdorff). In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... For quotient spaces in linear algebra, see quotient space (linear algebra). ...

Kernels in category theory

The notion of kernel in category theory is a generalisation of the kernels of abelian algebras; see Kernel (category theory). The categorical generalisation of the kernel as a congruence relation is the kernel pair. (There is also the notion of difference kernel, or binary equaliser.) Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ... This article is about equalisers in mathematics. ... This article is about equalisers in mathematics. ...